Eilenberg–Moore spectral sequence

Remarks

The proof compares the cochains of $E_f$ with the derived tensor product of the cochains of $X$ and $E$ over the cochains of $B$. The spectral sequence comes from filtering a bar construction by bar length. Some condition on $\pi_1(B)$ is needed for convergence; for general non-simply-connected $B$ the spectral sequence may fail to converge to the cohomology of the pullback.

Taking $X$ a point gives a spectral sequence $\operatorname{Tor}_{H^*(B)}(k, H^*(E)) \Rightarrow H^*(F)$ for the fibre $F$. Taking $X$ a point and $E$ the contractible path space gives $\operatorname{Tor}_{H^*(B)}(k, k) \Rightarrow H^*(\Omega B)$, the cohomology of the loop space.

Example: for $B = S^n$ with $n \ge 2$, $H^*(S^n; k)$ is an exterior algebra on a class $x$ of degree $n$. Then $\operatorname{Tor}_{H^*(S^n)}(k, k)$ has one copy of $k$ in each bidegree $(-j, jn)$, of total degree $j(n-1)$, and the spectral sequence collapses for degree reasons. This gives $H^i(\Omega S^n; k) \cong k$ when $n - 1$ divides $i$, and $0$ otherwise.

Going deeper

For a student

The spectral sequence is a key tool for computing the cohomology of loop spaces and of homogeneous spaces $G/H$ of Lie groups. It shows, for instance, that the cohomology of $\Omega B$ depends only on the cochain algebra of $B$ up to quasi-isomorphism when $B$ is simply connected. It is dual in spirit to the Serre spectral sequence, and the two are often used together.